Geographically weighted regression

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Geographically weighted regression

• Danlin Yu
• Yehua Dennis Wei
• Dept. of Geog., UWM

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Outline of the presentation

• Spatial non-stationarity an example
• GWR some definitions
• 6 good reasons using GWR
• Calibration and tests of GWR
• An example housing hedonic model in Milwaukee
• Further information

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1. Stationary v.s non-stationary
yi ?i0 ?i1x1i
yi ?0 ?1x1i
?e1
?e1
?e2
?e2
Stationary process
Non-stationary process
?e4
?e3
?e3
?e4
Assumed
More realistic
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Simpsons paradox
Spatially aggregated data
Spatially disaggregated data
House Price
House density
House density
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Stationary v.s. non-stationary

• If non-stationarity is modeled by stationary
  models
• Possible wrong conclusions might be drawn
• Residuals of the model might be highly spatial
  autocorrelated

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Why do relationships vary spatially?

• Sampling variation
• Nuisance variation, not real spatial
  non-stationarity
• Relationships intrinsically different across
  space
• Real spatial non-stationarity
• Model misspecification
• Can significant local variations be removed?

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2. Some definitions

• Spatial non-stationarity the same stimulus
  provokes a different response in different parts
  of the study region
• Global models statements about processes which
  are assumed to be stationary and as such are
  location independent

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Some definitions

• Local models spatial decompositions of global
  models, the results of local models are location
  dependent a characteristic we usually
  anticipate from geographic (spatial) data

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Regression

• Regression establishes relationship among a
  dependent variable and a set of independent
  variable(s)
• A typical linear regression model looks like
• yi?0 ?1x1i ?2x2i ?nxni?i
• With yi the dependent variable, xji (j from 1 to
  n) the set of independent variables, and ?i the
  residual, all at location i

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Regression

• When applied to spatial data, as can be seen, it
  assumes a stationary spatial process
• The same stimulus provokes the same response in
  all parts of the study region
• Highly untenable for spatial process

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Geographically weighted regression

• Local statistical technique to analyze spatial
  variations in relationships
• Spatial non-stationarity is assumed and will be
  tested
• Based on the First Law of Geography everything
  is related with everything else, but closer
  things are more related

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GWR

• Addresses the non-stationarity directly
• Allows the relationships to vary over space,
  i.e., ?s do not need to be everywhere the same
• This is the essence of GWR, in the linear form
• yi?i0 ?i1x1i ?i2x2i ?inxni?i
• Instead of remaining the same everywhere, ?s now
  vary in terms of locations (i)

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3. 6 good reasons why using GWR

• GWR is part of a growing trend in GIS towards
  local analysis
• Local statistics are spatial disaggregations of
  global ones
• Local analysis intends to understand the spatial
  data in more detail

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Global v.s. local statistics

• Global statistics
• Similarity across space
• Single-valued statistics
• Not mappable
• GIS unfriendly
• Search for regularities
• aspatial

• Local statistics
• Difference across space
• Multi-valued statistics
• Mappable
• GIS friendly
• Search for exceptions
• spatial

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6 good reasons why using GWR

• Provides useful link to GIS
• GISs are very useful for the storage,
  manipulation and display of spatial data
• Analytical functions are not fully developed
• In some cases the link between GIS and spatial
  analysis has been a step backwards
• Better spatial analytical tools are called for to
  take advantage of GISs functions

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GWR and GIS

• An important catalyst for the better integration
  of GIS and spatial analysis has been the
  development of local spatial statistical
  techniques
• GWR is among the recently new developments of
  local spatial analytical techniques

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6 good reasons why using GWR

• GWR is widely applicable to almost any form of
  spatial data
• Spatial link between health and wealth
• Presence/absence of a disease
• Determinants of house values
• Regional development mechanisms
• Remote sensing

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6 good reasons why using GWR

• GWR is truly a spatial technique
• It uses geographic information as well as
  attribute information
• It employs a spatial weighting function with the
  assumption that near places are more similar than
  distant ones (geography matters)
• The outputs are location specific hence mappable
  for further analysis

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6 good reasons why using GWR

• Residuals from GWR are generally much lower and
  usually much less spatially dependent
• GWR models give much better fits to data, EVEN
  accounting for added model complexity and number
  of parameters (decrease in degrees of freedom)
• GWR residuals are usually much less spatially
  dependent

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6 good reasons why using GWR

• GWR as a spatial microscope
• Instead of determining an optimal bandwidth
  (nearest neighbors), they can be input a priori
• A series of bandwidths can be selected and the
  resulting parameter surface examined at different
  levels of smoothing (adjusting amplifying factor
  in a microscope)

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6 good reasons why using GWR

• GWR as a spatial microscope
• Different details will exhibit different spatial
  varying patterns, which enables the researchers
  to be more flexible in discovering interesting
  spatial patterns, examining theories, and
  determining further steps

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4. Calibration of GWR

• Local weighted least squares
• Weights are attached with locations
• Based on the First Law of Geography everything
  is related with everything else, but closer
  things are more related than remote ones

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Weighting schemes

• Determines weights
• Most schemes tend to be Gaussian or Gaussian-like
  reflecting the type of dependency found in most
  spatial processes
• It can be either Fixed or Adaptive
• Both schemes based on Gaussian or Gaussian-like
  functions are implemented in GWR3.0 and R

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Fixed weighting scheme
Weighting function
Bandwidth
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Problems of fixed schemes

• Might produce large estimate variances where data
  are sparse, while mask subtle local variations
  where data are dense
• In extreme condition, fixed schemes might not be
  able to calibrate in local areas where data are
  too sparse to satisfy the calibration
  requirements (observations must be more than
  parameters)

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Adaptive weighting schemes
Weighting function
Bandwidth
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Adaptive weighting schemes

• Adaptive schemes adjust itself according to the
  density of data
• Shorter bandwidths where data are dense and
  longer where sparse
• Finding nearest neighbors are one of the often
  used approaches

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Calibration

• Surprisingly, the results of GWR appear to be
  relatively insensitive to the choice of weighting
  functions as long as it is a continuous
  distance-based function (Gaussian or
  Gaussian-like functions)
• Whichever weighting function is used, however the
  result will be sensitive to the bandwidth(s)

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Calibration

• An optimal bandwidth (or nearest neighbors)
  satisfies either
• Least cross-validation (CV) score
• CV score the difference between observed value
  and the GWR calibrated value using the bandwidth
  or nearest neighbors
• Least Akaike Information Criterion (AIC)
• An information criterion, considers the added
  complexity of GWR models

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Tests

• Are GWR really better than OLS models?
• An ANOVA table test (done in GWR 3.0, R)
• The Akaike Information Criterion (AIC)
• Less the AIC, better the model
• Rule of thumbs a decrease of AIC of 3 is
  regarded as successful improvement

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Tests

• Are the coefficients really varying across space
• F-tests based on the variance of coefficients
• Monte Carlo tests random permutation of the data

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5. An example

• Housing hedonic model in Milwaukee
• Data MPROP 2004 3430 samples used
• Dependent variable the assessed value (price)
• Independent variables air conditioner, floor
  size, fire place, house age, number of bathrooms,
  soil and Impervious surface (remote sensing
  acquired)

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The global model
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The global model

• 62 of the dependent variables variation is
  explained
• All determinants are statistically significant
• Floor size is the largest positive determinant
  house age is the largest negative determinant
• Deteriorated environment condition (large portion
  of soilimpervious surface) has significant
  negative impact

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GWR run summary

• Number of nearest neighbors for calibration 176
  (adaptive scheme)
• AIC 76317.39 (global 81731.63)
• GWR performs better than global model

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GWR run non-stationarity check
Tests are based on variance of coefficients, all
independent variables vary significantly over
space
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General conclusions

• Except for floor size, the established
  relationship between house values and the
  predictors are not necessarily significant
  everywhere in the City
• Same amount of change in these attributes
  (ceteris paribus) will bring larger amount of
  change in house values for houses locate near the
  Lake than those farther away

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General conclusions

• In the northwest and central eastern part of the
  City, house ages and house values hold opposite
  relationship as the global model suggests
• This is where the original immigrants built their
  house, and historical values weight more than
  house ages negative impact on house values

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6. Interested Groups

• GWR 3.0 software package can be obtained from
  Professor Stewart Fotheringham stewart.fotheringha
  m_at_MAY.IE
• GWR R codes are available from Danlin Yu directly
  (danlinyu_at_uwm.edu)
• Any interested groups can contact either
  Professor Yehua Dennis Wei (weiy_at_uwm.edu) or me
  for further info.

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Interested Groups

• The book Geographically Weighted Regression the
  analysis of spatially varying relationships is
  HIGHLY recommended for anyone who are interested
  in applying GWR in their own problems

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Acknowledgement

• Parts of the contents in this workshop are from
  CSISS 2004 summer workshop Geographically
  Weighted Regression Associated Statistics
• Specific thanks go to Professors Stewart
  Fotheringham, Chris Brunsdon, Roger Bivand and
  Martin Charlton

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Thank you all

• Questions and comments